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Lesson: Boxed In and Wrapped Up

Contributed by: Engineering K-Ph.D. Program, Pratt School of Engineering, Duke University
photo of shopper in cereal aisle of grocery store
We're all familiar with cereal boxes like these, but are they the best way to package the products inside?


Students find the volume and surface area of a rectangular box (e.g., a cereal box), and then figure out how to convert that box into a new, cubical box having the same volume as the original. As they construct the new, cube-shaped box from the original box material, students discover that the cubical box has less surface area than the original, and thus, a cube is a more efficient way to package things. Students then consider why consumer goods generally aren't packaged in cube-shaped boxes, even though they would require less material to produce and ultimately, less waste to discard. To display their findings, each student designs and constructs a mobile that contains a duplicate of his or her original box, the new cube-shaped box of the same volume, the scraps that are left over from the original box, and pertinent calculations of the volumes and surface areas involved. The activities involved provide valuable experience in problem solving with spatial-visual relationships.

Engineering Connection

Engineering analysis or partial design

Students learn to think like packaging engineers while considering ways in which consumer goods are boxed. They must consider not only the most efficient designs, but also how those designs will be used by the public.


  1. Pre-Req Knowledge
  2. Learning Objectives
  3. Introduction/Motivation
  4. Background
  5. Associated Activities
  6. Lesson Closure
  7. Attachments
  8. Assessment
  9. Extensions

Grade Level: 7 (6-8) Lesson #: 1 of 1
Time Required: 3.5 hours
Lesson Dependency :None
Keywords: rectangular prism, cube, volume, surface area
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Related Curriculum

subject areas Geometry
activities New Boxes From Old
The Boxes Go Mobile

Educational Standards :    

  •   Common Core State Standards for Mathematics: Math
  •   International Technology and Engineering Educators Association: Technology
  •   North Carolina: Math
Does this curriculum meet my state's standards?       

Pre-Req Knowledge (Return to Contents)

  • Students should know how to determine both the surface area and volume of a rectangular prism.

Learning Objectives (Return to Contents)

  • Students will be able to determine the dimensions of a cube when given its volume.
  • Students will be able to assert that a cube has less surface area than a rectangular prism of the same volume, and then prove this assertion with examples.

Introduction/Motivation (Return to Contents)

About two weeks before conducting this lesson, explain to students that for an upcoming activity, they will each need to bring in two identical boxes. Information about the kinds of boxes they should bring is provided in the attached Letter for Parents, which should be sent home with students. Students will be curious to know what they are going to do with the boxes, and you can help maintain this curiosity by merely answering with something vague such as, "They're for a geometry activity."
Before beginning the activity itself, students should know how to determine both the surface area and volume of a rectangular prism. They should also have a clear understanding of cubes: in a cube all the dimensions are equal, so the volume of a cube is the length of any side raised to the third power, or cubed.
To introduce the activity, tell students that they are going to take one of their two, identical boxes, and cut it up and tape it back together to make a cube-shaped box that has the same volume as the original, rectangular box. Then point out that in order to create a cube-shaped box from a rectangular box, they will have to work backwards. For a cube of known volume, they will need to be able to figure out how to find its dimensions: if the volume of a cube is equal to its length cubed, the length of any side of a cube is equal to the cube root of its volume. Some simple examples will help illustrate this. Ask the class, for example, what the dimensions would be for a box with a volume of 8 cubic centimeters, or for a box whose volume is 27 cubic inches.
Then move on to some harder examples. What if the volume of the box were 21 cubic inches? If students have graphing calculators such as the TI-82 or TI-83, they can find cube roots easily using the MATH function key. If not, they will get some good practice with estimation and trial-and-error as they determine that the cube root of 21 is about 2.76. (When they make their own cube-shaped boxes, they will work in centimeters and millimeters, so cube roots need not be taken out beyond the nearest hundredth.)
Once students are clear on how to work these types of surface area and volume problems, they are ready to begin the Associated Activities, New Boxes From Old and The Boxes Go Mobile.

Lesson Background & Concepts for Teachers (Return to Contents)

Volume-to-surface area ratios are important aspects of many phenomena in the physical and natural sciences. For example, radiators are devices designed to contain lots of surface area over which to dissipate heat, using a relatively small volume of hot fluid flowing through the radiator. Similarly, a long, narrow ranch-style house will cost a lot more to heat in a cold climate than will a more cube-shaped cape-cod style house having the same volume and wall insulation. The ranch house has more wall and roof areas through which the interior heat can escape than does the cape-cod house. Likewise, the ranch house will have a lot more area exposed to the radiant heat of the sun in the summer, and cost more to keep cool by air conditioning than will the cape-cod house.
In our own bodies, materials move in and out of our cells constantly, passing through the cell membranes primarily by the slow process of diffusion. The surface area of the cell determines how much material can be moved back and forth, and the smaller the cell, the greater the relative amount of surface area it contains. That is why cells are generally very small, with 10 microns (one one-hundredth of a millimeter) in diameter being a fairly typical cell size. Very large cells are rare, because without special mechanisms they can't take in enough nutrients and rid themselves of wastes fast enough to support the activities going on inside those large volumes. One-celled organisms are thus small and their life processes are fairly simple. More complicated organisms, such as ourselves, are multi-celled. By keeping our cells small, they can be specialized to do different jobs and yet still be maintained by the available nutrients and waste removal systems.

Associated Activities (Return to Contents)

  • New Boxes From Old - Students find the volume and surface area of a rectangular box (e.g., a cereal box) and then, using the same box material, construct a new cube-shaped box having the same volume as the original box.
  • The Boxes Go Mobile - To display the results of the New Boxes From Old activity, students design and construct a mobile that contains a duplicate of the original box, the new cube-shaped box of the same volume, the scraps that are left over from the original box, and pertinent calculations of the volumes and surface areas involved.

Lesson Closure (Return to Contents)

Ask your students to share with the class their answers to the last question on the New Boxes From Old student pages. While the total areas of their scraps should equal the differences in surface areas of their two boxes, it is unlikely that they will actually be very close. Measurement inaccuracies, rounding, and the difficulties of cutting straight lines and right angles will all combine to make their answers not as closely matched as they ought to be.
Since we are all concerned about preserving natural resources, ask the class which type of packaging would use the least paper: selling pasta, cereal, crackers, and cake mix, etc., in rectangular boxes, or in cube-shaped boxes. They should notice by now that rectangular boxes can be downright wasteful. Ask them to look around at all the mobiles and note which types of boxes generated the most scraps relative to the sizes of the boxes. They should be able to notice that long, thin boxes, such as spaghetti boxes or toothpaste boxes, had more scraps left over than boxes that had some faces that were square or nearly square, such as a diskette box. See if they can summarize their observations in mathematical terms, e.g., "When the length-to-width and length-to-height ratios are close to one, there are fewer scraps than when one or both of these ratios is much greater (or less) than one."
Since a cube-shaped box uses less material, why don't companies sell cereal and other foods this way? Ask students to share their thoughts about this question. If they need help, ask them to imagine how they would arrange many boxes of food in the same cabinet. Wouldn't lots of items have to be two or three rows back in the cabinet, and wouldn't items be stacked in at least two layers? What if they wanted the box of cereal that was all the way in the back and on the bottom layer?
Then ask them to think about picking up that cube-shaped box and pouring some cereal out of it. Would they have to hold the box with two hands because the box is so wide? Would this be awkward? And would the box now need a special pouring spout in order to get the cereal into a bowl instead of all over the counter? (One major cereal company recently experimented with a milk carton-style package.)
Students might also realize that when a consumer walks down the cereal aisle of a grocery store, each cereal company wants the consumer to buy its type(s) of cereal. The companies, therefore, want nice, big areas on their boxes so they can attract the consumer's attention and advertise what's inside. A cube-shaped box, with its smaller area facing the consumer, might not be as eye-catching as the usual rectangular box.
Some foods, because of their particular shapes, require rectangular packages. Spaghetti and lasagna noodles, for example, would have to be cut short to fit into a one-pound, cube-shaped box. Otherwise, a cube-shaped box containing standard-length noodles (about 26 cm) would be quite large. Just for fun, you can have your students determine the number of spaghetti noodles such a box would hold, if its dimensions were equal to the length of a typical noodle. (The answer depends on whether the box is filled with thick or thin spaghetti. One group of students counted 812 noodles in a one-pound box of thin spaghetti, which means there would be about 23,500 noodles in the cube-shaped box. Of course, there would be fewer noodles in a box of thick spaghetti noodles.) It is also interesting to note how heavy such a cubical box of noodles would be (29 pounds) and to speculate on whether or not the thin cardboard used in pasta boxes would be strong enough to support this weight (not likely).
  • Ask students to determine the difference in surface areas of a rectangular box of given dimensions, and a cube-shaped box having the same volume as the rectangular box.
  • Provide a sketch of a consumer product packaged in a rectangular box, and include the dimensions of the box in the sketch. Ask students to sketch a cube-shaped box, including its dimensions, that has the same volume as the rectangular box.
  • Ask students to write a paragraph explaining why consumer goods packaged in cube-shaped boxes would use less packaging material than rectangular boxes containing the same product volumes. Each student should provide an example, including sketches of the boxes and their dimensions, to substantiate his or her explanation.

Lesson Extension Activities (Return to Contents)

Find out if there is a packaging factory near your school for a field trip. Students and teachers alike will be amazed to see all the steps involved in designing, printing, cutting out, and assembling the boxes used to hold a variety of consumer products. To see if there is a packaging company in your area, look in the yellow pages under Packaging.

Other Related Information (Return to Contents)

This lesson and its associated activities were originally published, in slightly modified form, by Duke University's Center for Inquiry Based Learning (CIBL). Please visit the website (Bad Link http://www.ciblearning.org/resource.exercise.boxes.php) for information about CIBL and other resources for K-12 science and math teachers.


Mary R. Hebrank, Project Writer and Consultant, Duke University


© 2004 by Engineering K-Ph.D. Program, Pratt School of Engineering, Duke University
including copyrighted works from other educational institutions and/or U.S. government agencies; all rights reserved.

Supporting Program (Return to Contents)

Engineering K-Ph.D. Program, Pratt School of Engineering, Duke University

Last Modified: April 23, 2014
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